Problem: Determine how many solutions exist for the system of equations. ${-5x+y = 1}$ ${10x-2y = -2}$
Convert both equations to slope-intercept form: ${-5x+y = 1}$ $-5x{+5x} + y = 1{+5x}$ $y = 1+5x$ ${y = 5x+1}$ ${10x-2y = -2}$ $10x{-10x} - 2y = -2{-10x}$ $-2y = -2-10x$ $y = 1+5x$ ${y = 5x+1}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = 5x+1}$ ${y = 5x+1}$ Both equations have the same slope and the same y-intercept, which means the lines would completely overlap. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ Since any solution of ${-5x+y = 1}$ is also a solution of ${10x-2y = -2}$, there are infinitely many solutions.